The aim of this paper is to consider the $p$-adic local invariant cycle theorem in the mixed characteristic case.

\r\n\r\nIn the first part of the paper, via case-by-case discussion, we construct the $p$-adic specialization map, and then write out the complete conjecture in $p$-adic case. We proved the theorem in good reduction and semistable reduction cases.

\r\n\r\nIn the second part of the paper, by using Berthelot, Esnault and R\\\"{u}lling's trace morphisms in [BER], we first prove the case of coherent cohomology, then we extend it to the Witt vector cohomology, and we then get a result on the Frobenius-stable part of the Witt vector cohomology, which corresponds the slope 0 part of the rigid cohomology, we then get the general $p$-adic local invariant cycle theorem.

\r\n\r\nWe also give another approach in the $H^0$ and $H^1$ cases in the general case.

\r\n\r\nIn the last part of the paper, based on Flach and Morin's work on the weight filtration in the $l$-adic case, we consider the $p$-adic analogous result (which, together with the $l$-adic's result, serves as a part to prove the compatibility of the Weil-etale cohomology with the Tamagawa number conjecture). This is a direct corollary of the local invariant cycle theorem by taking the weight filtration. And we also consider some typical examples that the weight filtration statement could be verified by direct computations.

", "doi": "10.7907/ZJ9C-WT04", "publication_date": "2012", "thesis_type": "phd", "thesis_year": "2012" }, { "id": "thesis:8863", "collection": "thesis", "collection_id": "8863", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05182015-134458833", "type": "thesis", "title": "On the Construction of Higher \u00e9tale Regulators", "author": [ { "family_name": "Fan", "given_name": "Sin Tsun Edward", "clpid": "Fan-Sin-Tsun-Edward" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We present three approaches to define the higher \u00e9tale regulator maps \u03a6We study two conjectures introduced by Flach and Morin in [FM18] for schemes over a perfect field of characteristic *p* > 0. The first conjecture relates a *p*-adic extension of the \u00e9tale motivic cohomology with compact support on general schemes introduced by Geisser in [Gei06] to rigid cohomology with compact support, and is proved here. The second, relates a *p*-adic Borel-Moore motivic homology with the dual of rigid cohomology with compact support, and is proved in the smooth case. For this, we also prove a generalization of the comparison theorem from rigid cohomology to overconvergent de Rham-Witt cohomology in [DLZ11].

Let M be a motive that is defined over a number field and admits an action of a finite dimensional semisimple Q-algebra T. David Burns and Matthias Flach formulated a conjecture, which depends on a choice of Z-order T in T, for the leading coefficient of the Taylor expansion at 0 of the T-equivariant L-function of M. For primes l outside a finite set we prove the l-primary part of this conjecture for the specific case where M is the trace zero part of the adjoint of H\u00b9(X\u2080(N)) for prime N and where T is the (commutative) integral Hecke algebra for cusp forms of weight 2 and the congruence group \u0393\u2080(N), thus providing one of the first nontrivial supporting examples for the conjecture in a geometric situation where T is not the maximal order of T.

\r\n \r\nWe also compare two Selmer groups, one of which appears in Bloch-Kato conjecture and the other a slight variant of what is defined by A. Wiles. A result on the Fontaine-Laffaille modules with coefficients in a local ring finite free over Z_{\u2113} is obtained.